List all posts inside a category

I am a little bit confused because I thought this would be a simple task but I fail while implementing. I found similar requests and howtos, but either there are to old and don’t fit into the latest codes or I don’t understand them. And I also don’t succeed to change the code by myself based on my little knowledge and using trial and error.

It is just the wish to get a list of all posts inside a specified category that shows only the titles which should be linked. In other words: if I add a category to the menu, I will see the whole posts, one after the other, if I chose this link in the navigation. So it should be done with a little modification of the category.php, in my case the one from twenty twelve, right?:

<?php
/**
 * The template for displaying Category pages
 *
 * Used to display archive-type pages for posts in a category.
 *
 * @link https://codex.www.ads-software.com/Template_Hierarchy
 *
 * @package WordPress
 * @subpackage Twenty_Twelve
 * @since Twenty Twelve 1.0
 */

get_header(); ?>

	<section id="primary" class="site-content">
		<div id="content" role="main">

		<?php if ( have_posts() ) : ?>
			<header class="archive-header">
				<h1 class="archive-title"><?php printf( __( 'Category Archives: %s', 'twentytwelve' ), '<span>' . single_cat_title( '', false ) . '</span>' ); ?></h1>

			<?php if ( category_description() ) : // Show an optional category description ?>
				<div class="archive-meta"><?php echo category_description(); ?></div>
			<?php endif; ?>
			</header><!-- .archive-header -->

			<?php
			/* Start the Loop */
			while ( have_posts() ) : the_post();

				/* Include the post format-specific template for the content. If you want to
				 * this in a child theme then include a file called called content-___.php
				 * (where ___ is the post format) and that will be used instead.
				 */
				get_template_part( 'content', get_post_format() );

			endwhile;

			twentytwelve_content_nav( 'nav-below' );
			?>

		<?php else : ?>
			<?php get_template_part( 'content', 'none' ); ?>
		<?php endif; ?>

		</div><!-- #content -->
	</section><!-- #primary -->

<?php get_sidebar(); ?>
<?php get_footer(); ?>

Another idea is to get such a list as a sub menu, when I hover the cursor over the category link in the nav menu. Is this possible to?

I researched now and found out that there are also plugins to create list-like pages but I am not sure what might be a better solution to reach my goal. That is why I finally decided to ask for suggestions and help here.

Bye
Jan